11 70 Analysis of Scientific Books and Memoirs. 



ly small constant thickness, perpendicular to a vertical longitudinal plane. 

 If we commence our consideration at the stern, and advance gradually for- 

 ward, it is evident that the sections comprising the counter and its con- 

 necting parts, being free from the water, will be subject to no reaction 

 from it ; and when at last any reaction does take place, it must at first, 

 from the peculiar form of the body, be infinitely less than the weight of 

 the section whose displacement occasions it. As we approach, however, 

 the midship section of the vessel, the upward section of the fluid will ap- 

 proach more and more to an equality with the weight of its corresponding 

 section, and ultimately become equal to it ; and if we pass beyond this 

 section, and which may be denominated the section of hydrostatic equili- 

 brium, we shall find the weight of the water displaced become greater 

 than the weight of the section above it. In like manner, if we commence 

 at the bow of the vessel, we shall find a similar section of hydrostatic equi- 

 librium, and afterwards a like increase of the weight of the water displaced 

 above the weight of the section reposing on it. 



Dupin, in his paper Sur la Structure des Vaisseaux Anglais, has given a 

 fine analytical view of the subject ; and has furnished the differential equa- 

 tions on which the whole problem of arching depends. Our limits will 

 not permit us to follow the analytical steps of this beautiful inquiry, and 

 we can only furnish the following theorems resulting from them. 



I. That when a vertical plane divides a vessel into two parts, so that the 

 weight of each part is equal to the v>eight of water displaced by it, the mo- 

 ments of those parts estimated in relation to the same plane, to produce what 

 is denominated arching, will either he a maximum or a minimum. 



II. That this effect will be a maximum when the infinitely small section, 

 contiguous to the plane of the moments has its own moment in a contrary direc- 

 tion to that of the total moment. 



III. That the effect will be a minimum when this section has its own 

 moment acting in the same direction as the total moment. 



These theorems are applied by Dupin to the distribution of the forces 

 operating on the hull of a seventy-four gun ship when fitted for sea ; the 

 numerical elements relating to the weights and displacements of the seve- 

 ral sections being derived from Young's paper contained in the Philoso- 

 phical Transactions for 1814. We regret exceedingly that we cannot lay 

 before our readers the whole of this highly interesting inquiry, but must 

 pass on to remark, that the causes of arching are not due entirely to the 

 unequal distribution of the weight and pressure, but that the longitudinal 

 and horizontal pressure of the water also contributes to this alteration of 

 form. Dr Young has remarked that the partial pressure of the water in 

 a longitudinal direction affects the lower part of the ship only, compres- 

 sing and shortening the keel, while it has no immediate action on the up- 

 per decks. The pressure thus applied must obviously occasion a curva- 

 ture if the angles made with the decks by the timbers are supposed to re- 

 main unaltered, while the keel is shortened in the same manner as any 

 soft and thick substance, pressed at one edge between the fingers will be- 

 come concave at the part compressed ; and this strain upon the most pro- 

 bable upposition respecting the comparative strength of the upper and 



